# Properties

 Degree $24$ Conductor $1.796\times 10^{36}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·4-s + 4·7-s + 21·16-s − 24·28-s + 8·37-s − 12·43-s + 22·49-s − 56·64-s + 64·67-s + 56·79-s − 4·81-s − 56·109-s + 84·112-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 48·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 30·169-s + 72·172-s + 173-s + ⋯
 L(s)  = 1 − 3·4-s + 1.51·7-s + 21/4·16-s − 4.53·28-s + 1.31·37-s − 1.82·43-s + 22/7·49-s − 7·64-s + 7.81·67-s + 6.30·79-s − 4/9·81-s − 5.36·109-s + 7.93·112-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.94·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.30·169-s + 5.48·172-s + 0.0760·173-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1050} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$9.806842285$$ $$L(\frac12)$$ $$\approx$$ $$9.806842285$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + T^{2} )^{6}$$
3 $$1 + 4 T^{4} - 10 p T^{6} + 4 p^{2} T^{8} + p^{6} T^{12}$$
5 $$1$$
7 $$( 1 - 2 T - 5 T^{2} + 36 T^{3} - 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
good11 $$( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 320 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
13 $$( 1 - 15 T^{2} + 43 p T^{4} - 5110 T^{6} + 43 p^{3} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
17 $$( 1 + 29 T^{2} + 494 T^{4} + 5297 T^{6} + 494 p^{2} T^{8} + 29 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
19 $$( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 4132 p^{2} T^{8} - 96 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
23 $$( 1 - 73 T^{2} + 105 p T^{4} - 58154 T^{6} + 105 p^{3} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
29 $$( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 5535 p^{2} T^{8} - 109 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
31 $$( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2623 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
37 $$( 1 - 2 T + p T^{2} - 300 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
41 $$( 1 - 3 T^{2} + 3166 T^{4} - 23 p T^{6} + 3166 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
43 $$( 1 + 3 T + 109 T^{2} + 266 T^{3} + 109 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
47 $$( 1 + 78 T^{2} + 7315 T^{4} + 322124 T^{6} + 7315 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
53 $$( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 13615 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
59 $$( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 18587 p^{2} T^{8} + 179 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
61 $$( 1 - 3 p T^{2} + 12451 T^{4} - 609898 T^{6} + 12451 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} )^{2}$$
67 $$( 1 - 16 T + 258 T^{2} - 2108 T^{3} + 258 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
71 $$( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 21667 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
73 $$( 1 - 332 T^{2} + 52356 T^{4} - 4849846 T^{6} + 52356 p^{2} T^{8} - 332 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
79 $$( 1 - 14 T + 193 T^{2} - 2172 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
83 $$( 1 + 309 T^{2} + 44338 T^{4} + 4238477 T^{6} + 44338 p^{2} T^{8} + 309 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
89 $$( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 58036 p^{2} T^{8} + 348 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
97 $$( 1 - 158 T^{2} + 21375 T^{4} - 2986564 T^{6} + 21375 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$